A NONCOMMUTATIVE LIMIT-THEOREM FOR HOMOGENEOUS CORRELATIONS

We state and prove a noncommutative limit theorem for correlations whi ch are homogeneous with respect to order-preserving injections. The mo st interesting examples of central limit theorems in quantum probabili ty (for commuting, anticommuting, and free independence and also vario us q-qclt's), as well as the limit theorems for the Poisson law and th e free Poisson law are special cases of the theorem. In particular, th e theorem contains the q-central limit theorem for non-identically dis tributed variables, derived in our previous work in the context of q-b ialgebras and quantum groups. More importantly, new examples of limit theorems of q-Poisson type are derived for both the infinite tensor pr oduct algebra and the reduced free product, leading to new q-laws. In the first case the limit as q --> 1 is studied in more detail and a co nnection with partial Bell polynomials is established.